Coordinate system

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For an elementary introduction to this topic, see coordinates (mathematics)
The Cartesian coordinate system.

In mathematics and applications, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. "Numbers" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. If the space or manifold is curved, it may not be possible to provide one consistent coordinate system for the entire space. In this case, a set of coordinate systems, called charts, are put together to form an atlas covering the whole space.

When the space has some additional algebraic structure, then the coordinates will also transform under rings or groups; a particularly famous example in this case are the Lie groups.

Although any specific coordinate system is useful for numerical calculations in a given space, the space itself is considered to exist independently of any particular choice of coordinates. By convention the origin of the coordinate system in Cartesian coordinates is the point (0, 0, ..., 0), which may be assigned to any given point of Euclidean space.

In some coordinate systems some points are associated with multiple tuples of coordinates, e.g. the origin in polar coordinates: r = 0 but θ can be any angle.


[edit] Examples

An example of a coordinate system is to describe a point P in the Euclidean space Rn by an n-tuple

P = (r1, ..., rn)

of real numbers

r1, ..., rn.

These numbers r1, ..., rn are called the coordinates of the point P.

If a subset S of a Euclidean space is mapped continuously onto another topological space, this defines coordinates in the image of S. That can be called a parametrization of the image, since it assigns numbers to points. That correspondence is unique only if the mapping is bijective.

The system of assigning longitude and latitude to geographical locations is a coordinate system. In this case the parametrization fails to be unique at the north and south poles.

[edit] Transformations

A coordinate transformation is a conversion from one system to another, to describe the same space.

With every bijection from the space to itself two coordinate transformations can be associated:

  • such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
  • such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)

For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more.

[edit] Systems commonly used

Some coordinate systems are the following:

[edit] Astronomical systems

[edit] Less common coordinate systems

The following coordinate systems have special uses. They all have the properties of being orthogonal coordinate systems, that is the coordinate surface meet at right angles.

[edit] See also

ca:Coordenada cs:Soustava souřadnic da:Koordinatsystem de:Koordinatensystem et:Koordinaatsüsteem es:Sistema de coordenadas eo:Koordinatsistemo fr:Système de coordonnées gl:Sistema de coordenadas ko:좌표계 it:Sistema di riferimento he:קואורדינטות lb:Koordinate-system lt:Koordinačių sistema nl:Coördinaat ja:座標 no:Koordinatsystem pl:Układ współrzędnych pt:Sistema de coordenadas ru:Система координат sk:Sústava súradníc sv:Koordinatsystem ta:பகுமுறை வடிவியல் uk:Cистеми координат в елементарній математиці zh:座標系統

Coordinate system

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